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Full text of "Treatise On Analysis Vol-Ii"

394       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

for each function u e *%C(K), there exists a unique continuous operator T(u) on E such
that

for all xt y in E. In order that u\~*T(u) should be a representation of ^C(K) on E, it
is necessary and sufficient that

(x | y)

for all x, y in E and all u e ^C(K). In the last relation, the condition u e
be replaced by u e ^C(K).

2.    (a)   Deduce from Problem 1 a new proof of the fact that a representation u i  T(u) of
^C(K) in a Hilbert space E can be uniquely extended to a representation u f > jT()
of ^c(K) on E, satisfying the relation (15.10.4.1).   (To prove that T(uv) = T(u)T(v)
for H, z; e ^C(K), proceed in two steps :   first assume that one of the two functions is
continuous and use the relation fcr(>  x, y = ^X.TCJO . y when w is continuous; then prove
that this relation remains valid for all u e <^C(K).)

(b)    If a continuous operator Ke J2?(E) commutes with T(u) for all w e ^C(K), then
it commutes with T(v) for all y e ^C(K).

(c)   If E is separable, show that each operator T(u), where u e ^C(K), is the limit, with
respect to the weak topology on &(E) (Section 12.15, Problem 9) of a sequence
(T(vn)), where vne&c(l); and conversely. (For the converse, use (15.10.6(ii)) and
(b) above.)

3.    With the notation of Problem 1, show that T()  x = 0 for x e E and u e ^C(K) if
and only if u is //.*, ^-negligible for all y in a total subset of E. The mapping wh-*T(w)
of ^c(K) into JS?(E) is injective if and only if the union of the supports of the measures
jit,, y is dense in K.

4.    With the notation of Problem 1 , suppose that E is separable; let (a/) be a dense sequence
in E, and put vtj = | p,ait aj\. Then there exists a positive measure v on K such that the
relation v(N) = 0 is equivalent to the relation " vu(N) = 0 for all i and j " (13.15.8).
All the functions in WC(K.) which are equivalent with respect to v have the same image
under T, and we may therefore regard T as an injective homomorphism of Lg(v) into
^(E). Prove that N(w) g ll^()ll for all functions u e ^g(v) (consider first the case
where u is a step-function). Consequently T is a bicontinuous mapping of Lg(*>) onto
its image in

5.   Suppose that, under the hypotheses of (15.10.8) there exist two sequences (En,/xn)
and (E, /4) satisfying the conditions of (15.10.9).

(a) Show that if p,n = 0 for n ^ 2, then f4 =* 0 for n 2> 2. (In other words, a topologic-
ally cyclic representation does not admit a canonical decomposition satisfying (15.10.9)
as the sum of at least two topologically cyclic representations.) To prove this, observe
that if a is a totalizing vector for T, and a'n is its projection on E'n , then a'n ^ 0 if E'n ^ 0
(i.e., if [Mfn ^ 0). If afn = 0 for some n ^ 2, show that there would exist a sequence (gm)
of functions belonging to ^C(K), such that T(gm)  a( tends to 0 and T(gm)  a'n tends to
a'n, and show that this contradicts the fact that p,'H is a measure with base fc-i . /*